Welcome to the website of researchers working in Bilbao (either at UPV/EHU or at BCAM) in mathematical analysis, partial differential equations, inverse problems, mathematical physics, and/or related topics.
BCAM, Thursday, October 16th, 2025, 17:00 - 18:00
Title: Convex integration for the Monge-Ampere system.
Marta Lewicka - University of Pittsburgh
The Monge-Ampere system (MA) is the multi-dimensional version of the Monge-Ampere equation, arising from the prescribed curvature problem and closely related to the problems of isometric immersions and the minimization of elastic energies of thin shells.
In case of dimension d=2 and codimension k=1, (MA) reduces to the classical Monge-Ampere equation as the prescription of the Gaussian curvature of a shallow surface in R^3, whereas (MA) prescribes the full Riemann curvature of a shallow d-dimensional manifold in R^{d+k}.
(MA) takes its weak formulation, called the Von Karman system (VK). When d= 2, k= 1, (VK) arises in the theory of elasticity as the the Von Karman stretching content a thin film.
Closely related to (MA) and (VK) is the system (II) for an isometric immersion of the given d-dimensional Riemannian metric into R^{d+k}. (II) yields (VK) when equating the leading order terms along a perturbation of the Euclidean metric.
This lecture will concern the ongoing study of existence, regularity, and multiplicity of solutions to systems (MA), (VK), (II) through the method of convex integration, building on the prior fundamental results due to Nash, Kuiper, Kallen, Borisov, the more recent approach due to Conti, Delellis and Szekelyhidi, and the parallel analysis of Cao, Hirsch and Inauen.
We will also explore relation to the scaling of the non-Euclidean energies of elastic deformations and the quantitative isometric immersion problem.